Spherical harmonics

Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the value of Yℓm(θ,ϕ){\displaystyle Y_{\ell }^{m}(\theta ,\phi )} in angular direction (θ,ϕ){\displaystyle (\theta ,\phi )}.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree ℓ{\displaystyle \ell } in (x,y,z){\displaystyle (x,y,z)} that obey Laplace's equation. Functions that satisfy Laplace's equation are often said to be harmonic, hence the name spherical harmonics. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of rℓ{\displaystyle r^{\ell }} from the above-mentioned polynomial of degree ℓ{\displaystyle \ell }; the remaining factor can be regarded as a function of the spherical angular coordinates θ{\displaystyle \theta } and φ{\displaystyle \varphi } only, or equivalently of the orientational unit vector r{\displaystyle {\mathbf {r} }} specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition.

A specific set of spherical harmonics, denoted Yℓm(θ,φ){\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} or Yℓm(r){\displaystyle Y_{\ell }^{m}({\mathbf {r} })}, are called Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782.[1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.

Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. He discovered that if r ≤ r1 then

where γ is the angle between the vectors x and x1. The functions Pi are the Legendre polynomials, and they are a special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x1 and x. (See Applications of Legendre polynomials in physics for a more detailed analysis.)

By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre.

The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.

Real (Laplace) spherical harmonics Yℓm for ℓ = 0, …, 4 (top to bottom) and m = 0, …, ℓ (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics Yℓ−m{\displaystyle Y_{\ell }^{-m}} would be shown rotated about the z axis by 90∘/m{\displaystyle 90^{\circ }/m} with respect to the positive order ones.)

The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations

for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e± i m φ. The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ (ℓ + 1) for some non-negative integer with ℓ ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomialPℓm(cos θ) . Finally, the equation for R has solutions of the form R(r) = A rℓ + B r−ℓ − 1; requiring the solution to be regular throughout R3 forces B = 0.[3]

Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

Here Yℓm is called a spherical harmonic function of degree ℓ and order m, Pℓm is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitudeθ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitudeφ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ, φ) of the eigenvalue problem

r2∇2Y=−ℓ(ℓ+1)Y{\displaystyle r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y}

is a linear combination of Yℓm. In fact, for any such solution, rℓ Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2ℓ + 1 linearly independent such polynomials.

The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor rℓ,

Thus L+ : Eλ,m → Eλ,m+1 (it is a "raising operator") and L− : Eλ,m → Eλ,m−1 (it is a "lowering operator"). In particular, Lk
+ : Eλ,m → Eλ,m+k must be zero for k sufficiently large, because the inequality λ ≥ m2 must hold in each of the nontrivial joint eigenspaces. Let Y ∈ Eλ,m be a nonzero joint eigenfunction, and let k be the least integer such that

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of (−1)m for m > 0, 1 otherwise, commonly referred to as the Condon–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. There is no requirement to use the Condon–Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy[7] and magnetics communities never include the Condon–Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.[citation needed]

The real spherical harmonics are sometimes known as tesseral spherical harmonics.[8] These functions have the same orthonormality properties as the complex ones above. The harmonics with m > 0 are said to be of cosine type, and those with m < 0 of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as

As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span the same space as the complex ones would.

For example, as can be seen from the table of spherical harmonics, the usual p functions (l=1{\displaystyle l=1}) are complex and mix axis directions, but the real versions are essentially just x, y and z.

Essentially all the properties of the spherical harmonics can be derived from this generating function.[9] An immediate benefit of this definition is that if the c-number vector r{\displaystyle {\mathbf {r} }} is replaced by the quantum mechanical spin vector operator J{\displaystyle {\mathbf {J} }}, one obtains a generating function for a standardized set of spherical tensor operators, Yℓm(J){\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })}:

The parallelism of the two definitions ensures that the Yℓm{\displaystyle {\mathcal {Y}}_{\ell }^{m}}'s transform under rotations (see below) in the same way as the Yℓm{\displaystyle Y_{\ell }^{m}}'s, which in turn guarantees that they are spherical tensor operators, Tq(k){\displaystyle T_{q}^{(k)}}, with k=ℓ{\displaystyle k={\ell }} and q=m{\displaystyle q=m}, obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.

The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of z{\displaystyle z} and another of x{\displaystyle x} and y{\displaystyle y}, as follows (Condon-Shortley phase):

Using the equations above to form the real spherical harmonics, it is seen that for m>0{\displaystyle m>0} only the Am{\displaystyle A_{m}} terms (cosines) are included, and for m<0{\displaystyle m<0} only the Bm{\displaystyle B_{m}} terms (sines) are included:

The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator PΨ(r)=Ψ(−r){\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )}. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with r{\displaystyle \mathbf {r} } being a unit vector,

In terms of the spherical angles, parity transforms a point with coordinates {θ,ϕ}{\displaystyle \{\theta ,\phi \}} to {π−θ,π+ϕ}{\displaystyle \{\pi -\theta ,\pi +\phi \}}. The statement of the parity of spherical harmonics is then

(This can be seen as follows: The associated Legendre polynomials gives (−1)ℓ+m and from the exponential function we have (−1)m, giving together for the spherical harmonics a parity of (−1)ℓ.)

Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree ℓ changes the sign by a factor of (−1)ℓ.

Consider a rotation R{\displaystyle {\mathcal {R}}} about the origin that sends the unit vector r{\displaystyle \mathbf {r} } to r′{\displaystyle {\mathbf {r} }'}. Under this operation, a spherical harmonic of degree ℓ{\displaystyle \ell } and order m{\displaystyle m} transforms into a linear combination of spherical harmonics of the same degree. That is,

where Amm′{\displaystyle A_{mm'}} is a matrix of order (2ℓ+1){\displaystyle (2\ell +1)} that depends on the rotation R{\displaystyle {\mathcal {R}}}. However, this is not the standard way of expressing this property. In the standard way one writes,

where Dmm′(ℓ)(R)∗{\displaystyle D_{mm'}^{(\ell )}({\mathcal {R}})^{*}} is the complex conjugate of an element of the Wigner D-matrix.

The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The Yℓm{\displaystyle Y_{\ell }^{m}}'s of degree ℓ{\displaystyle \ell } provide a basis set of functions for the irreducible representation of the group SO(3) of dimension (2ℓ+1){\displaystyle (2\ell +1)}. Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.

The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:

The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics):

is defined as the cross-power spectrum. If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then Sff(ℓ) and Sfg(ℓ) represent the contributions to the function's variance and covariance for degree ℓ, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form

Sff(ℓ)=Cℓβ.{\displaystyle S_{f\!f}(\ell )=C\,\ell ^{\beta }.}

When β = 0, the spectrum is "white" as each degree possesses equal power. When β < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when β > 0, the spectrum is termed "blue". The condition on the order of growth of Sff(ℓ) is related to the order of differentiability of f in the next section.

The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the Sff(ℓ) to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if

where Pℓ is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics.[11] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side.[12]

In the expansion (1), the left-hand side Pℓ(x·y) is a constant multiple of the degree ℓ zonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let Yj be an arbitrary orthonormal basis of the space Hℓ of degree ℓ spherical harmonics on the n-sphere. Then Zx(ℓ){\displaystyle Z_{\mathbf {x} }^{(\ell )}}, the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as[14]

The Clebsch–Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics itself. A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. Abstractly, the Clebsch–Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.

Schematic representation of Yℓm{\displaystyle Y_{\ell m}} on the unit sphere and its nodal lines. Re[Yℓm]{\displaystyle {\text{Re}}[Y_{\ell m}]} is equal to 0 along mgreat circles passing through the poles, and along ℓ−m circles of equal latitude. The function changes sign each time it crosses one of these lines.

The Laplace spherical harmonics Yℓm{\displaystyle Y_{\ell }^{m}} can be visualized by considering their "nodal lines", that is, the set of points on the sphere where Re[Yℓm]=0{\displaystyle {\text{Re}}[Y_{\ell }^{m}]=0}, or alternatively where Im[Yℓm]=0{\displaystyle {\text{Im}}[Y_{\ell }^{m}]=0}. Nodal lines of Yℓm{\displaystyle Y_{\ell }^{m}} are composed of circles: some are latitudes and others are longitudes. One can determine the number of nodal lines of each type by counting the number of zeros of Yℓm{\displaystyle Y_{\ell }^{m}} in the latitudinal and longitudinal directions independently. For the latitudinal direction,[clarification needed] the real and imaginary components of the associated Legendre polynomials each possess ℓ−|m| zeros, whereas for the longitudinal direction, the trigonometric sin and cos functions possess 2|m| zeros.[clarification needed]

When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. When ℓ = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.

More general spherical harmonics of degree ℓ are not necessarily those of the Laplace basis Yℓm{\displaystyle Y_{\ell }^{m}}, and their nodal sets can be of a fairly general kind.[15]

The classical spherical harmonics are defined as functions on the unit sphere S2 inside three-dimensional Euclidean space. Spherical harmonics can be generalized to higher-dimensional Euclidean space Rn as follows.[16] Let Pℓ denote the space of homogeneous polynomials of degree ℓ in n variables. That is, a polynomial P is in Pℓ provided that

The sum of the spaces Hℓ is dense in the set of continuous functions on Sn−1 with respect to the uniform topology, by the Stone-Weierstrass theorem. As a result, the sum of these spaces is also dense in the space L2(Sn−1) of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series a spherical harmonics, where the series converges in the L2 sense.

It follows from the Stokes theorem and the preceding property that the spaces Hℓ are orthogonal with respect to the inner product from L2(Sn−1). That is to say,

∫Sn−1fg¯dΩ=0{\displaystyle \int _{S^{n-1}}f{\bar {g}}\,d\Omega =0}

for f ∈ Hℓ and g ∈ Hk for k ≠ ℓ.

Conversely, the spaces Hℓ are precisely the eigenspaces of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potentialΔSn−1−1{\displaystyle \Delta _{S^{n-1}}^{-1}} gives another proof that the spaces Hℓ are pairwise orthogonal and complete in L2(Sn−1).

Every homogeneous polynomial P ∈ Pℓ can be uniquely written in the form

An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian

The space Hℓ of spherical harmonics of degree ℓ is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on Hℓ by function composition

The elements of Hℓ arise as the restrictions to the sphere of elements of Aℓ: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R3. By polarization of ψ ∈ Aℓ, there are coefficients ψi1…iℓ{\displaystyle \psi _{i_{1}\dots i_{\ell }}} symmetric on the indices, uniquely determined by the requirement

The condition that ψ be harmonic is equivalent to the assertion that the tensorψi1…iℓ{\displaystyle \psi _{i_{1}\dots i_{\ell }}} must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), Hℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ.

More generally, the analogous statements hold in higher dimensions: the space Hℓ of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓ-tensors. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner.

The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.